Interest rate and stock return volatility indices for the Eurozone. Investors gauges of fear during the recent financial crisis *

Transcription

1 Interest rate and stock return volatlty ndces for the Eurozone. Investors gauges of fear durng the recent fnancal crss * Raquel López a, Elseo Navarro b Abstract We suggest a methodology for the constructon of a set of nterest rate volatlty ndces for the Eurozone (s) based on the mpled volatlty quotes of caps (floors), one of the most lqud nterest rate dervatves. These ndces reflect the market s aggregate expectaton of volatlty of forward rates over both short- and long-term horzons (from one to ten years ahead). Volatlty ndces n equty markets are referred to as nvestors gauges of fear because they usually spke n perods of market turmol. In ths paper, we extend the emprcal evdence by analyzng the effect of the recent fnancal crss on short- and long-term s. We fnd that the level of short-term s (70%) as of Aprl 2012 s stll far from returnng to the average pre-crss value (17%) and that the crss has also affected nvestors long-term expectatons of volatlty. In addton, usng two stock return volatlty ndces for the Eurozone, we fnd that the crss has had a deeper mpact on nvestors uncertanty about the evoluton of nterest rates than on stock market returns. Keywords: caps and floors, crss, nterest rates; nvestors gauge of fear, volatlty ndces * Raquel and Elseo acknowledge the fnancal support provded by Mnstero de Educacón y Cenca grant ECO Any remanng errors are our responsblty alone. a Raquel López s n the Department of Economc Analyss and Fnance, Unversty of Castlla-La Mancha (E-mal: raquel.lopez@uclm.es). b Elseo Navarro s n the Department of Busness Scence, Unversty of Alcalá (E-mal: elseo.navarro@uah.es).

2 1. Introducton The number of mpled volatlty ndces has sgnfcantly ncreased over the last decade n the equty markets all over the world (e.g., VIX for US; VDAX, VCAC and VSTOXX for Europe; VKOSPI and Inda VIX for Asa; S&P/ASX 200 VIX for Australa; and New SAVI for South Afrca). These ndces capture the market s expectaton of volatlty of stock ndces returns over a very short-term horzon (usually 30 calendar days) and have a number of nterestng applcatons. They have been used for forecastng future realzed volatlty (see e.g., Moraux et al., 1999; Bluhm and Yu, 2001; Becker et al., 2007), to assess the underlyng stock ndces market rsk (Got, 2005a), to dentfy proftable opportuntes n the stock market (see e.g., Got, 2005b; Banerjee et al., 2007), and to measure fnancal rsk averson (see e.g., Barros et al., 2009; Beber et al., 2009; Gerlach et al., 2010; Arghyrou and Kontonkas, 2011; Olvera et al., 2012). In ths study we suggest a methodology for the constructon of a set of nterest rate volatlty ndces for the Eurozone (s), whch reflect the market estmate of the volatlty of three- and sx-month tenor forward rates over dfferent fxed horzons one, two, three, four, fve, seven and ten years. To the best of our knowledge, there are no volatlty ndces calculated by exchanges or other nsttutons for the Eurozone fxed-ncome market. Thus, ths paper contrbutes to the lterature on volatlty ndces by coverng ths gap. To ths end, we use data on caps (floors), one of the most lqud over-the-counter (OTC) nterest rate dervatve contracts (L and Zhao, 2009). Bank for Internatonal Settlements statstcs, as of December 2011, ndcate that nterest rate optons (caps, floors, collars and corrdors) were the second most-traded OTC nterest rate dervatves worldwde. Moreover, the notonal amount of OTC nterest rate optons exceeded that of exchange-traded optons by nearly $20 trllon. Sorted by currency, Euro nterest rate optons accounted for approxmately 46% of the total amount outstandng of OTC nterest rate optons traded n the world. 1 Usng daly data from January 2004 to Aprl 2012, ths paper analyzes how short- and long-term expectatons of nterest rate volatlty change n response to the fnancal turmol durng the present crss. Concernng short-term s, we observe large spkes along upward and downward slopes snce the summer of As of Aprl 2012, 1 1

3 volatlty levels of approxmately 70% are stll far from returnng to the average precrss value (17%). Thus, the crss has had a deep and lastng effect on nvestors shortterm expectatons of volatlty n the fxed-ncome market. More nterestngly, we also fnd that as the crss deepened, t also eventually affected expectatons of volatlty fve and ten years ahead: the ndces ntate an upward trend n In addton, we compare the effect of the crss on nvestors uncertanty about the future development of nterest rates and stock market returns usng two VSTOXX volatlty sub-ndces constructed from Dow Jones EURO STOXX 50 optons exprng n one and two years. In the equty markets, VIX s usually called the nvestors gauge of fear because t spkes durng perods of market turmol (Whaley, 2000, 2009). In ths paper, we observe that VSTOXX sub-ndces exhbt a lower rse than one- and two-year s along the crss perod and that the sze of the spkes s also smaller. Ths fndng suggests that s have played a greater role as nvestors gauge of fear durng the recent fnancal crss than VSTOXX sub-ndces. Based on these fndngs, we construct a measure of global fnancal rsk averson usng stock return and nterest rate volatlty ndces and we show that ths ndcator of rsk averson and Euro area government bond yeld spreads move closely together from 2009 up to md Fnally, we prove that movements n short-term s and VSTOXX sub-ndces are postvely correlated and that changes n the nterest rate volatlty ndex mght be useful for portfolo managers to mprove stock return volatlty forecasts. The rest of the paper s organzed as follows. The next secton examnes caps (floors) valuaton accordng to the Lbor Market Model (LMM), whch s consstent wth the market standard approach for prcng these contracts usng the Black prcng formula. In Secton Three, we present the methodology for the calculaton of s. Secton Four descrbes the database. In Secton Fve, we analyze the behavor and statstcal propertes of s, and compare the effect of the fnancal turmol durng the recent crss on the market estmates of future volatlty of nterest rates and stock returns. Fnally, Secton Sx ncludes the conclusons of the study. 2

4 2. Caps and floors valuaton. The LMM and the Black formula Caps and floors are portfolos of optons on nterest rates, caplets and floorlets. Thus, the desgn and valuaton of caps (floors) can best be understood by frst descrbng the optons that comprse them. Caplets (floorlets) are European-style call (put) optons where the underlyng asset s a forward rate agreement (FRA). An FRA s an agreement between two partes to exchange an amount of money proportonal to the dfference between the fxed strke rate K (set at t) and the floatng nterest rate (reset at tme T ) whch prevals over the perod [T,T +τ], L(T,T +τ), ( t T < T + τ ). The payoff of an FRA at T +τ s: NP [ L T, T + τ K ] τ (, (1) ) where NP s the notonal prncpal of the contract and τ s the tenor nterval. Caplets (floorlets) are exercsed only f L(T,T +τ) s greater (smaller) than the strke K. The payoff of a caplet at T +τ s: { L( T, T + τ ) K, } τ NP Max 0, (2) and the payoff of a floorlet s: { K L T, T +τ ), } τ NP Max 0. (3) ( The LMM assumes that the forward nterest rate f(t,t,t +τ) follows a lognormal stochastc process (see Brgo and Mercuro, 2006 for an extensve revew of LMM). Takng nto account that the lmtng value of the forward rate when t approaches T s equal to the floatng nterest rate L(T,T +τ), and assumng there are no arbtrage opportuntes, the well-known Black (1976) prcng formulas for valung caplets (floorlets) are derved (see e.g., Díaz et al., 2009): [ ] Capl ettt (,, τ, K, σ ) = NP f(, tt, T+ τ) Nh ( ) K Nh ( ) PtT (, + τ) τ, (4) K, Black 1 2 [ ] Floor lettt (,, τ, K, σ ) = NP K N( h) f(, tt, T+ τ) N( h) PtT (, + τ) τ,(5) K, Black 2 1 3

5 where h h 1 2 ln = ln = [ f ( t, T, T + τ ) / K ] σ K, Black [ f ( t, T, T + τ ) / K ] σ K, Black 1 K + ( σ, 2 ( T t) 1 K ( σ, 2 ( T t) Black Black ) ) 2 2 ( T ( T t), (6) t). (7) K K Caplet t, T, τ, K, σ ) and Floorlet t, T, τ, K, σ ) are the prces at t of a caplet (, Black (, Black and a floorlet, respectvely, T s the exercse date of the opton (and the maturty date of the underlyng forward rate), τ s the tenor of the underlyng forward rate (and T +τ s the maturty date of the opton), P(t,T +τ) s the prce at t of a unt-zero coupon bond wth maturty at T +τ, N( ) s the cumulatve normal dstrbuton, and s the socalled Black mpled volatlty of an opton wth exercse date T and strke K. Black mpled volatlty can be understood, wthn the LMM, as an average of the nstantaneous volatlty of the log of the forward rate f(t,t,t +τ) over the perod [t,t ]: T 2 σ ( u, T ) du K 2 t ( σ, Black ) =, (8) ( T t) where s the nstantaneous volatlty at t of the lognormal process followed by the forward rate f(t,t,t +τ). Caps (floors) are portfolos of caplets (floorlets) wth the same strke and tenor but wth consecutve maturtes so that the maturty date of each caplet (floorlet) concdes wth the exercse date of the followng one. In the Eurozone, caps (floors) wth tme to expraton up to two years have a three-month tenor, whereas the tenor for caps (floors) wth maturtes beyond two years s sx months. Thus, a two-year cap (floor) conssts of a chan of seven caplets (floorlets) wth exercse dates n three, sx, nne, 12, 15, 18 and 21 months, whereas a three-year cap (floor) comprses fve caplets (floorlets) wth exercse dates n sx, 12, 18, 24 and 30 months. Please note that, unlke equty optons, caplets (floorlets) and caps (floors) have a constant lfe perod. 4

6 The payoffs generated by a cap (floor) can be descrbed as follows. On the exercse date of the frst caplet (floorlet), the floatng rate s observed and compared to the strke. If the floatng rate s greater (smaller) than the strke, then on the second reset date the seller of the cap (floor) pays the holder the dfference between the floatng rate (strke) and the strke (floatng rate) multpled by the notonal prncpal and the tenor. If the floatng rate s less (more) than the strke, there s no payoff from the cap (floor). Thus, through the lfe of a cap (floor), payments are due at the end of each tenor nterval, although the amount s known at the reset date (at the begnnng of the tenor nterval) when the floatng nterest rate s observed. 2 Then, the prce at tme t of an n-year cap wth strke K can be obtaned as the sum of the values of the caplets that comprse t. That s, n k 1 = 1 K Cap( t, Tn k, K) = caplet( t, T, τ, K, σ, Black ), (9) where k equals 4 (2) when the tenor nterval s three (sx) months, and T 1, T 2,...,T n k-1 are the reset dates of the cap that concde wth the exercse dates of the caplets that compose the cap and T n k = T n k-1 +τ,.e., the date that the last cash flow wll be due f L(T n k-1,t n k-1 +τ) > K. An analogous formula can be set up for the prce of a floor. However, quotatons n the cap market are computed assumng that the volatlty of all the caplets that compose a partcular cap s the same. In fact, an n-year cap wth strke K s quoted by the market through the so-called flat volatlty, whch s the constant value that equals the sum of the values of all the caplets that compose the cap accordng to the Black formula to ts market prce,.e., the value such that n k 1 = 1 K Cap( t, Tn k, K) = caplet( t, T, τ, K, σ n, Black ). 3 (10) Therefore, flat volatltes cannot be consdered to be a pure measure of the future evoluton of volatlty of a forward rate; rather, they are a mxture of the average future 2 Caps (floors) are usually defned so that the ntal floatng rate, even f t s greater (smaller) than the strke, does not lead to a payoff on the frst reset date (Hull, 2009). 3 Actually, the market quotes flat volatltes of caps/floors. At a partcular strke and for a concrete term to maturty, traders may contract the same nstrument as a cap or a floor dependng on ther expectatons. 5

7 volatltes of a set of forward rates wth consecutve terms to maturty. 4 Thus, for nstance, the flat volatlty of a two-year cap s a mxture of the average future volatlty of three-month tenor forward rates wth maturtes n three, sx, nne, 12, 15, 18 and 21 months. Fnally, note that accordng to the LMM and Equaton (8), the mpled volatlty of caplets should be the same for all caplets wth the same term to maturty, ndependent of the strke K. However, n practce, the mpled volatlty of caplets and caps (wth everythng else equal) vares wth the strke rate, gvng rse to volatlty surfaces ( see Jarrow et al., 2007). 3. Methodology We develop a set of nterest rate volatlty ndces that capture the market s expected volatlty of a partcular forward rate over dfferent fxed horzons usng the mpled volatlty quotes of caps (floors). 5,6 However, the use of data from ths market poses the problem of havng to address a contract where the underlyng rate s not a sngle forward rate but a set of forward rates wth consecutve maturtes. Therefore, the constructon of s nvolves recoverng the mpled volatltes of the ndvdual caplets that compose caps usng a strppng procedure ( see e.g., Hernández, 2005). Ths process conssts of obtanng the prce at tme t of a caplet wth a strke K and reset date T, caplet(t,t,τ,k, ), by subtractng the prces of two consecutve caps wth the same strke K: 4 The dfference between and s smlar to the dfference between zero-coupon rates and the yelds to maturty of coupon-bearng bonds. 5 Stock return volatlty ndces are calculated usng the market prces of (exchange-traded) optons, rather than ther respectve mpled volatltes, based on the concept of the far delvery value of future realzed varance suggested by Demeterf et al. (1999). However, note that the quoted opton prce n the OTC market s actually mpled volatlty tself (.e., mpled volatlty does not need to be nferred from opton prces). Thus, to provde an mpled volatlty quote n the cap (floor) market means to gve the opton prce, smlar to how the yeld to maturty of a bond s an alternatve way of provdng the prce of the bond. 6 Impled volatltes of specfc forward rates could be drectly obtaned from caplet (floorlet) quotatons, however, these contracts are qute llqud; thus, obtanng a complete enough range of caplets (floorlets) wth dfferent maturtes can be complcated. Thus, the constructon of s from caps (floors) data can gve a much more accurate ndcaton of the actual uncertanty regardng the future behavor of nterest rates for a wde range of maturtes, wthout the ntruson of the nose caused by the lack of lqudty. 6

8 K Caplet t, T, τ, K, σ ) Cap( t, T, K) Cap( t, T, ), (11) (, Black = + 1 K where Cap( ) are defned as n Equaton (10). Once the prce of the caplet s obtaned, the Black prcng formula s used to derve the correspondng mpled volatlty. 7 Note that when mplementng the strppng procedure, the Black s model s used only to translate volatlty quotes nto opton prces and vce versa. Thus, we are not makng use of any of the assumptons of the Black s model. It s merely used as a tool to provde a one-to-one mappng between opton prces and mpled volatltes. As stressed n the prevous secton, t must be noted that we obtan dfferent mpled Black volatltes for the same caplet, dependng upon the strke rate K. Thus, a decson on the strkes of the caps used for the mplementaton of s needs to be made. Poon and Granger (2003) suggest usng at-the-money (ATM) optons because they are more lqud and less prone to measurement errors. In the cap market, an n-year cap s sad to be ATM f the strke of ths nstrument equals the fxed rate of a swap that has the same payment days as the cap (see e.g., Hull, 2009). However, we cannot use ATM caps n the strppng process because two consecutve caps would have dfferent strkes to the extent that swaps wth dfferent maturtes usually have dfferent fxed rates. Therefore, we must address the problem of determnng the strke of an ATM caplet. Accordng to the Black formula, a caplet s sad to be ATM when the value of the underlyng forward rate equals the strke rate. Thus, we propose usng the avalable caps wth strkes closest to the outstandng forward rate f(t,t,t +τ) defned as P( t, T ) 1 f ( t, T, + τ ) = 1 (, τ ) T, (12) P t T + τ where P(t,T ) and P(t,T +τ) are the prces at t of unt zero-coupon bonds wth maturtes at T and T +τ, respectvely. In partcular, we wll use caps wth strkes mmedately above and below f(t,t, T +τ), and we wll refer to them as K A and K B, respectvely, wth K B < f(t,t,t +τ) < K A. Then, usng Equaton (11), we obtan the prces of caplets wth strkes K A (frst out-ofthe-money caplet) and K B (frst n-the-money caplet), and we derve ther mpled 7 Note that the same mpled volatlty s obtaned when the prcng formulas for floors (floorlets) are used nstead. 7

9 volatltes usng the Black formula. We denote these two mpled volatltes by and, respectvely. Fnally, we use lnear nterpolaton to obtan : t T A B K K f ( t, T, T + τ ) A K f ( t, T, T + τ K σ + ) B, Black A B σ, (13) Black A K K K K (, ) =, B where (t,t ) s the annualzed mpled volatlty of a theoretcal ATM caplet wth a constant tme to maturty, from t to T. Accordng to Carr and Lee (2003, 2009) the volatlty swap rate wth expry at tme T s well approxmated by the ATM mpled volatlty maturng at the same tme. A volatlty swap s a contract traded OTC that pays at maturty the dfference between the realzed volatlty of the underlyng asset over the lfe of the contract and a fxed volatlty rate (the volatlty swap rate). Snce the contract has zero value at the tme of entry, by no arbtrage, the volatlty swap rate equals the condtonal rsk-neutral expected value of the realzed volatlty over the lfe of the contract. Thus, (t,t ) approxmates the condtonal rsk-neutral expectaton of the realzed volatlty of the underlyng forward nterest rate f(t,t,t +τ) over the perod [t, T ]. In addton, because s based on the market quotes of very lqud optons, t represents a consensus market vew of the expected volatlty of the underlyng forward rate. Usng the constructon method just descrbed, we create a daly set of nterest rate volatlty ndces for three- (sx-) month tenor forward rates exprng n one and two (three, four, fve, seven and ten) years. Accordng to Duarte et al. (2005), these are the most lqud cap maturtes. Accordng to Equaton (8), each provdes the average future volatlty of a forward nterest rate up to ts maturty. For nstance, (t,1y) measures the market s assessment at any tme t of the uncertanty regardng the evoluton of the forward rate f(t,t+1y,t+1y+3m) over the next year; and (t,10y) would ndcate the average volatlty of the forward rate f(t,t+10y,t+10y+6m) over the next ten years. Thus, unlke flat volatltes, s measure the volatlty of specfc forward rates. For nstance, σ K 1, flat would be some sort of average of the future volatltes of the 8

10 forward rates f(t,t+3m,t+6m), f(t,t+6m,t+9m) and f(t,t+9m,t+1y) up to ther respectve maturtes. 4. Data For the constructon of s we use two sets of daly data from the Eurozone fxedncome market. The frst set conssts of last bd-ask averages flat volatlty quotes of caps (floors) for a fxed set of maturtes and strkes retreved from Bloomberg. The data suppler for these quotes s the large OTC nterdealer broker ICAP. The second set conssts of zero-coupon curves provded by Reuters based on the most lqud rate nstruments avalable, a combnaton of deposts, lqud futures and nterest rate swaps. The sample extends from January 02, 2004 to Aprl 30, Flat volatltes correspond to caps (floors) wth maturtes of one to ten plus 12, 15 and 20 years and wth the followng range of strke rates: 0.01, 0.02, , 0.025, 0.03, 0.04, 0.05, 0.06 and These strkes cover the range of values of the forward rates durng the sample perod to ensure that there wll be always a strke above and below the outstandng forward rates. 9 Note also that the two strkes closest to the forward rate f(t,t,t +τ) can dffer by only 25, 50 or 100 bass ponts. Thus, the assumpton that the volatlty smle s well approxmated by a lne that we use when the mpled volatltes of near-the-money optons are lnearly nterpolated for the constructon of s consdered reasonable due to the small range of strkes over whch the nterpolaton s made ( Flemng et al., 1995). Note also that the strppng procedure nvolves usng the prces of caps wth maturtes n one (two) years and three months, and three (four, fve, seven and ten) years and sx months, whereas markets only provde caps wth annual terms to maturty (.e., wth an nteger number of years to maturty). Therefore, nterpolaton and extrapolaton technques must be used to obtan flat volatltes of caps wth a maturty dfferent from those quoted. Interpolaton and extrapolaton technques are appled between caps wth the same tenor nterval. Thus, when the requred maturty cap s below three years, we apply lnear 8 The sample starts n January 2004 because of data avalablty. 9 The only excepton occurs for the forward rate maturng n one year snce values below one percent are observed snce md Thus, n ths partcular stuaton, s just the mpled volatlty of the caplet (floorlet) wth a strke rate of

11 nterpolaton/extrapolaton by usng the one- and two-year maturtes. For the rest of maturtes, we apply cubc splne nterpolaton based on the flat volatltes of caps wth maturtes of three to ten years plus 12, 15 and 20 years (see Hernández, 2005). We use lnear nterpolaton/extrapolaton only when the number of avalable flat volatltes s less than sx. Note that these nterpolaton/extrapolaton technques must produce unquely determned values of unobservable flat volatltes wth any term to maturty up to ten years and sx months (the maturty date of the caplet wth an exercse date n ten years). See the Appendx for a detaled descrpton of the nterpolaton/extrapolaton procedure. In regards to measurng the market s expectatons of volatlty n the equty market, we use two volatlty ndces dstrbuted by STOXX Ltd, VSTOXX 12M and VSTOXX 24M. Actually, they belong to the set of sub-ndces that are calculated n addton to the man ndex, VSTOXX, whch measures volatlty over the next 30 calendar days. In partcular VSTOXX 12M and VSTOXX 24M are constructed based on the prces of Dow Jones EURO STOXX 50 optons exprng n 12 and 24 months, respectvely. Thus, they capture the market s expected volatlty of the Dow Jones EURO STOXX 50 returns over the next 12 and 24 months. Sub-ndces based on optons wth longer terms to expraton are not currently avalable Emprcal analyss In ths secton we analyze the behavor and statstcal propertes of the set of s. Then, we analyze the role of nterest rate and stock return volatlty ndces as nvestors gauges of fear durng the crss Propertes of s The daly evoluton of s wth tmes to maturty of one, two, fve and ten years from January 02, 2004 to Aprl 30, 2012 s shown n Fgure 1. We observe a decreasng pattern n s wth the closest forecast horzons from the begnnng of the sample up to approxmately md-july Then, the ndces ntate an upward trend whch leaves market estmates of nterest rate volatlty over the next one and two years at approxmately 70% n May Thus, short-term s seem 10 Addtonal nformaton on the ndces can be found on 10

12 to reflect the fnancal turmol snce the begnnng of the crss. By May 2010, the upward trend turns a downward trend untl approxmately Aprl 2011, when the level of s s close to 30%. Then, s exhbt a new outstandng rse that drves market expectatons of nterest rate volatlty to a maxmum of approxmately 90% n fve months. Fnally, large (up and down) spkes are observed along a downward slope untl Aprl 2012, when s level (approxmately 70%) s stll far from returnng to the average pre-crss value. Fgure 1. Daly levels of (t,1y), (t,2y), (t,5y) and (t,10y) over the perod from January 02, 2004 to Aprl 30, % 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% (t,1y) (t,2y) (t,5y) (t,10y) Concernng nvestors expectatons of volatlty over the next fve and ten years, they exhbt a rse durng the frst half of 2010 and approxmately double by the end of the sample. Thus, two man conclusons can be drawn from the graphcs. On the one hand, the fact that the ndces (especally the short-term ones) spke and sharply ncrease durng the recent fnancal crss supports the nterpretaton of as a gauge of fear for fxedncome markets smlar to the wdely held vew of VIX for equty markets (Whaley, 11

13 2000, 2009). On the other hand, the fact that long-term s also respond to the fnancal turmol seems to suggest that nvestors foresee long perods of turbulence n nterest rate markets. Recall that provdes the average level of future volatlty untl the maturty of the underlyng forward rate (see Equaton (8)), and hence, only a lastng shft n the market estmates of future volatlty would make long-term s rase. The summary statstcs of the set of nterest rate volatlty ndces for the full sample as well as before and after the begnnng of the subprme crss are ncluded n Table 1 (Panels A, B and C, respectvely). The frst subsample spans the perod from January 02, 2004 to July 31, 2007 (885 observatons), and the second spans the perod from August 01, 2007 to Aprl 30, 2012 (1177 observatons). 11 The average value of all s ncreases durng the crss perod. In addton, we fnd that the mean of all maturty s s qute smlar before the crss, whereas t progressvely decays as the forecast horzon ncreases for the second subsample. Shortterm s also show greater varablty (standard devaton) than long-term ones before and durng the crss. The skewness and kurtoss measures suggest that the ndces are closer to a normal dstrbuton n the splt sample than when the whole sample s consdered. In any case, the Jarque-Bera test does not accept the null hypothess of a normal dstrbuton for any of the ndces n any of the two subperods. To nvestgate whether the seres are statonary, the augmented Dckey-Fuller (ADF) unt root test for ts most general specfcaton (.e., wth ntercept and lnear trend) s performed on the logarthm of the volatlty ndces. The null hypothess of a unt root s not rejected n any case August 2007 s usually referred to as the onset date of the subprme crss (see e.g., Taylor and Wllams, 2009; Flemng and Klagge, 2010), and a change n the values of the ndces s also especally perceptble around ths date. 12 Gven the lkely exstence of a structural break n the seres durng the crss perod, the modfed verson of the ADF test developed by Zvot and Andrews (2002) to allow for a structural break n the data s conducted. The null hypothess s that the seres follows a unt root process; the alternatve hypothess mples that the seres s a trend-statonary process wth a one-tme break n the trend functon occurrng at an unknown pont n tme. We obtan that the null hypothess contnues not beng rejected n all the cases, except for (t,4y) at the 5% sgnfcance level. Moreover, we fnd that the structural break dates dentfed by the test for short-term s belong to July

14 Table 1. Summary statstcs of s across the entre sample (Panel A) and for two subsamples: from January 02, 2004 to July 31, 2007 (Panel B) and from August 01, 2007 to Aprl 30, 2012 (Panel C) (t,1y) (t,2y) (t,3y) (t,4y) (t,5y) (t,7y) (t,10y) Panel A: January 02, 2004 to Aprl 30, 2012 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera ρ ** 0.99 ** 0.99 ** 0.99 ** 0.99 ** 0.98 ** 0.97 ** ADF Panel B: January 02, 2004 to July 31, 2007 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera Panel C: August 01, 2007 to Aprl 30, 2012 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera Notes: a p-values of the Jarque-Bera test are nsde parenthess. b ρ 1 denotes the frst-order autocorrelaton coeffcent. The sgnfcance of autocorrelatons s tested wth the Ljung-Box Q-statstc. c The ADF test s performed on the logarthm of the ndces. The null hypothess s that the seres contans a unt root. The optmal lag length s determned accordng to the Schwarz nformaton crteron. d One and two astersks denote statstcal sgnfcance at the 5% and 1% sgnfcance level, respectvely. Summary statstcs for the daly log-dfferences of s are also shown n Table 2. On the one hand, the excess kurtoss found n the seres s also reported by Dotss et al. 13

15 (2007) for several equty market volatlty ndces n ther frst dfferences, where the non-normalty may be attrbuted to the presence of jumps n mpled volatlty. On the other hand, the sgnfcant negatve frst-order autocorrelaton supports the modelng of mpled volatlty ndces as mean-revertng processes. All the seres are statonary after dfferencng. To formally nvestgate whether there are statstcally sgnfcant dfferences n the dstrbuton of the ndces before and after the begnnng of the crss we apply two nonparametrc tests. Panel A n Table 3 shows the results of the Wlcoxon/Mann-Whtney test for the equalty of medans and the Brown-Forsythe test for the equalty of varances for the seres n levels. The results show evdence of sgnfcant dfferences n both the medan and the varance between the frst and second subsamples at the 1% sgnfcance level for all the ndces. For the seres n frst log-dfferences (Panel B n Table 3), statstcally sgnfcant dfferences n the medans between the frst and second subsamples are unproven for all forecast horzons; however, the null hypothess of equalty of varances s rejected for s wth tme to expraton from one to fve years. 14

16 Table 2. Summary statstcs of frst log-dfferences of s across the entre sample (Panel A) and for two subsamples: from January 02, 2004 to July 31, 2007 (Panel B) and from August 01, 2007 to Aprl 30, 2012 (Panel C) (t,1y) (t,2y) (t,3y) (t,4y) (t,5y) (t,7y) (t,10y) Panel A: January 02, 2004 to June 30, 2011 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera ρ ** ** ** ** ** ** ** ADF ** ** ** ** ** ** ** Panel B: January 02, 2004 to July 31, 2007 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera Panel C: August 01, 2007 to Aprl 30, 2012 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera Notes: a p-values of the Jarque-Bera test are nsde parenthess. b ρ 1 denotes the frst-order autocorrelaton coeffcent. The sgnfcance of autocorrelatons s tested wth the Ljung-Box Q-statstc. c The ADF test s performed on the logarthm of the ndces. The null hypothess s that the seres contans a unt root. The optmal lag length s determned accordng to the Schwarz nformaton crteron. d One and two astersks denote statstcal sgnfcance at the 5% and 1% sgnfcance level, respectvely. 15

17 Table 3. Tests of equalty of medans and varances between the frst and second subsamples for the ndces n levels (Panel A) and n frst log-dfferences (Panel B) (t,1y) (t,2y) (t,3y) (t,4y) (t,5y) Panel A: Tests of equalty of medans and varances for s n levels (t,7y) (t,10y) Wlcoxon/Mann- Whtney test ** ** ** ** ** ** ** Brown-Forsythe test ** ** ** ** ** ** ** Panel B: Tests of equalty of medans and varances for s n frst log-dfferences Wlcoxon/Mann- Whtney test Brown-Forsythe test ** ** ** ** ** Notes: The null hypothess of the Wlcoxon/Mann-Whtney test s that the medans are equal. The null hypothess of the Brown-Forsythe test s that the varances are equal. One and two astersks denote rejecton of the null hypothess at the 5% and 1% sgnfcance level, respectvely The role of nterest rate and stock return volatlty ndces as nvestors gauges of fear durng the fnancal crss In ths secton we use (t,1y) and (t,2y) along wth VSTOXX 12M and VSTOXX 24M to track how nvestors uncertanty about the future behavor of nterest rates and stock returns one and two years ahead changes n response to fnancal nstablty durng the recent fnancal crss. Fgure 2 plots the four mentoned ndces across the sample. Smlar to s, we can also see an ncrease n the VSTOXX sub-ndces by the summer of However, the sze of the spkes observed n the VSTOXX seres along the crss perod s notably smaller than n the case of s. The standard devaton of VSTOXX 12M s 6% (Table 4), whereas t s 17% for (t,1y). The hghest volatlty level (49.73%) s reached by VSTOXX 12M on November 21, Also note that for approxmately one year before and after the burst of the crss, market estmates of future volatlty are hgher n the equty market than n the fxed-ncome market. However, by March 2009, VSTOXX 12M and VSTOXX 24M decrease and from that moment untl the end of the sample the stock return volatlty ndces reman below s. Moreover, by Aprl 2012, VSTOXX sub-ndces are approxmately 30%, whereas the average pre-crss level of the ndces s approxmately 20%. Smlar fndngs are documented by Schwert 16

18 (2011) for the US stock market based on VIX. He shows that the recent fnancal crss has had the second-largest burst of volatlty after the market crash n October 1987, although t seems that stock volatlty returned to more normal levels farly quckly. Fgure 2. Daly levels of (t,1y), (t,2y), VSTOXX 12M and VSTOXX 24M over the perod from January 02, 2004 to Aprl 30, % 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% (t,1y) (t,2y) VSTOXX 12M VSTOXX 24M Two deas may be nferred from the comparson of stock return and nterest rate volatlty ndces. Frst, we fnd that the fnancal turmol has had a deeper mpact on nvestors uncertanty about the future development of nterest rates than on stock market returns. Put dfferently, s have played a greater role as nvestors gauge of fear durng the recent fnancal crss than VSTOXX sub-ndces. Second, t s nterestng to note that both fxed-ncome and equty market volatlty ndces show a qute smlar pattern over most of the sample except from March 2009 to Aprl 2010, when VSTOXX sub-ndces start to fall whle s keep an upward trend. Based on these fndngs, next we further nvestgate two ssues. 17

19 Table 4. Summary statstcs of VSTOXX 12M and VSTOXX 24M n levels and frst log-dfferences across the entre sample (Panel A) and for two subsamples: from January 01, 2004 to July 31, 2007 (Panel B) and from August 01, 2007 to Aprl 30, 2012 (Panel C) Levels Frst log-dfferences VSTOXX 12M VSTOXX 24M VSTOXX 12M VSTOXX 24M Panel A: January 01, 2004 to Aprl 30, 2012 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera ρ ** 0.99 ** ** ** ADF ** ** Panel B: January 01, 2004 to July 31, 2007 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera Panel C: August 01, 2007 to Aprl 30, 2012 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera Notes: a p-values of the Jarque-Bera test are nsde parenthess. b ρ 1 denotes the frst-order autocorrelaton coeffcent. The sgnfcance of autocorrelatons s tested wth the Ljung-Box Q-statstc. c The ADF test s performed on the logarthm of the ndces. The null hypothess s that the seres contans a unt root. The optmal lag length s determned accordng to the Schwarz nformaton crteron. d One and two astersks denote statstcal sgnfcance at the 5% and 1% sgnfcance level, respectvely. 18

20 Frst, we construct a measure of fnancal rsk averson as a lnear combnaton of stock return and nterest rate volatlty ndces and we analyze ts relatonshp wth the development of Euro area government bond yeld spreads durng the crss. Second, we analyze whether there s a transmsson of mpled volatlty across the Eurozone fxedncome and equty markets based on s and VSTOXX sub-ndces. The emprcal evdence on mpled volatlty contagon across equty markets has ncreased n the recent years (see e.g., Nkknen and Sahlström, 2004; Äjö, 2008; Konstantnd et al., 2008; Kumar, 2012; Sropoulos and Fassas, 2012), whereas the correlaton between stock return and nterest rate volatlty ndces remans unexplored. These results have mplcatons for portfolo management Fnancal rsk averson and Euro area government yeld spreads Gven the nvestors gauge of fear role of volatlty ndces, VIX and VSTOXX have been extensvely used as proxes for nvestors rsk averson n recent studes on Euro area government yeld spreads (see e.g., Barros et al., 2009; Beber et al., 2009; Gerlach et al., 2010; Arghyrou and Kontonkas, 2011; Olvera et al., 2012). Partcularly, Arghyrou and Kontonkas (2011) and Olvera et al. (2012) show that the coeffcents of VIX and VSTOXX, respectvely, are postve and statstcally sgnfcant for explanng the wdenng of soveregn yeld spreads n the present crss. The ratonale behnd these fndngs can be found n De Sants (2012). Hgher rsk averson durng the crss has ncreased the demand for German bonds for safety reasons, whle the yelds of the bonds wth hgher default rsk has ncreased. Based on the hgher percepton of uncertanty n the fxed-ncome markets over most of the crss perod, we expect that short-term s contans sgnfcant nformaton about the development of Euro area government yeld spreads. The prevous ssue s nvestgated usng prncpal component analyss. Prncpal component analyss allows us to decompose the behavor of a set of varables closely related nto orthogonal components, the frst of whch captures the common factor behnd the behavor of the varables. Specfcally, each component s derved as a lnear combnaton of the orgnal data n such a way that they are n descendng order of contrbuton for explanng the total varablty of the set of varables. Thus, we use ths method to analyze the relatonshp between the common soveregn rsk factor derved from a seres of Euro area government bond yeld spreads and a measure of global 19

21 fnancal rsk averson obtaned from stock return and nterest rate volatlty ndces durng the crss perod (from August 01, 2007 to Aprl 30, 2012). See Barros et al. (2009) for a related approach. In partcular, the common soveregn rsk factor s the frst prncpal component of the yeld spreads between ten-year Euro area government bonds and the benchmark German bond. Daly data have been obtaned from Reuters for the followng lst of countres: Belgum, Fnland, France, Greece, Ireland, Italy, the Netherlands, Portugal and Span. 13 Wth respect to the fnancal rsk averson factor, ths s the frst prncpal component of a set of four volatlty ndces: two stock return volatlty ndces (VIX and VSTOXX) and two nterest rate volatlty ndces (MOVE and (t,1y). MOVE s computed by Merrll Lynch as a weghted average of the mpled volatltes of one-month ATM optons on the two-year, fve-year, ten-year and 30-year US Treasury securtes. The data for VIX has been collected from the webste of the Chcago Board of Optons Exchange (CBOE), whereas the data for MOVE has been obtaned from Bloomberg. Pror to analyss, each seres s standardzed to have zero mean and unt varance by subtractng the mean and dvdng by the standard devaton n each case. Panel A (Panel B) n Table 5 shows the weghts of each yeld spread (volatlty ndex) for each of the nne (four) prncpal components. The frst prncpal component of the yeld spread seres explans 74% of the varaton of spreads, whereas the frst prncpal component of the volatlty ndces captures 54% of the total varaton n the ndces. The loadngs on each yeld spread makng up the common soveregn rsk factor are qute smlar across countres, although the weghts of Fnland and the Netherlands are lower. The loadngs on the four volatlty ndces for the rsk averson factor are all postve and show that US volatlty ndces contrbute to the common factor to a greater extent than Euro area volatlty ndces. The second component, however, whch explans 37% of the varablty of volatlty ndces, only places postve weghts on Euro area volatlty ndces. Ths result seems to hghlght the hgher and more prolonged rsk averson wthn the Eurozone as the fnancal crss deepens n comparson wth the US. 13 The remanng countres are not ncluded because of data avalablty or because they joned the euro after August In partcular, Austra s excluded from the sample because of mssng data from August to the end of September

22 As dsplayed n Fgure 3, global fnancal rsk averson and Euro area government bond yeld spreads move qute close from 2009 up to md In partcular, the generalzed drop n our measure of nvestors rsk averson translates nto lower soveregn rsk. However, the behavor of the two varables notably dverges snce the summer of 2011, when the soveregn rsk factor sharply ncreases whle the perceved rsk n fnancal markets s reduced. Durng ths perod, Greece s debt crss threatened to spread to the bgger economes of Italy and Span. In ths respect, Olvera et al. (2012) hghlght the sgnfcant roles that macroeconomc country-specfc fundamentals such as the level of publc debt and the current account defct have played for explanng the rse n soveregn rsk durng the crss. Table 5. Weghts of yeld spreads and volatlty ndces for each prncpal component (Panels A and B, respectvely). Prncpal components are n descendng order of contrbuton for explanng the total varablty of the set of varables. The data extends from August 01, 2007 to Aprl 30, 2012 Component number Panel A: Weghts of yeld spreads for each prncpal component Belgum Fnland France Greece Ireland Italy Netherlands Portugal Span Panel B: Weghts of volatlty ndces for each prncpal component (t,1y) MOVE VIX VSTOXX

23 Fgure 3. Fnancal rsk averson and soveregn rsk ndcators from August 01, 2007 to Aprl 30, Soveregn rsk factor Fnancal rsk averson Spllover effects Volatlty contagon across markets has mportant mplcatons for portfolo choce and rsk management because t affects opton prces and hedge ratos. Moreover, f correlatons across markets rse n perods of stress, then dversfcaton benefts wll be reduced. In ths secton, we analyze whether there s a transmsson of mpled volatlty across the Eurozone fxed-ncome and equty markets based on s and VSTOXX sub-ndces. Table 6 shows the cross-correlatons between weekly log-changes n the four ndces for the entre sample (Panel A) as well as for the pre-crss and crss perods (Panels B and C, respectvely). We fnd that there s a statstcally sgnfcant postve correlaton between changes n nterest rate and stock return volatlty ndces over the whole sample, although ths s stronger durng the crss perod - the hghest correlaton s obtaned for (t,1y) and VSTOXX 12M (31%). Debold and Ylmaz (2009) also fnd that volatlty spllovers across stock markets become more pronounced durng perods of crses. Fndngs have mplcatons for opton portfolo managers because they suggest a postve relatonshp between the change n the prce of caps (floors) and the change n the prce of Dow Jones EURO STOXX 50 optons. 22

24 Next, we test whether there s a Granger causalty relatonshp between changes n (t,1y) and VSTOXX 12M. To ths end, we estmate the followng bvarate vector autoregressve (VAR) model: k k l ( t,1 Y ) = a + b l ( t l,1 Y l) + c lvstoxx12m + u, (14) l l t l t l= 1 l= 1 k k lvstoxx12m = a + b lvstoxx12 M + c l ( t l,1 Y l) + u, (15) t l t l l t l= 1 l= 1 where l ( t,1 Y ) and lvstoxx12m t stand for the frst log-dfferences of (t,1y) and VSTOXX 12M, respectvely, and k denotes the number of lags used n the regresson. The null hypothess s that the c l n Equatons (14) and (15) are jontly zero. Thus, VSTOXX changes Granger-cause changes f lagged VSTOXX changes contan nformaton that s not already contaned n the past values of. Granger causalty runnng from to VSTOXX can be defned n a smlar way. Table 6. Cross-correlatons between weekly log-changes n (t,1y), (t,2y), VSTOXX 12M and VSTOXX 24M across the entre sample (Panel A) and for two subsamples: from January 02, 2004 to July 31, 2007 (Panel B) and from August 01, 2007 to Aprl 30, 2012 (Panel C) Cross-correlatons Panel A: January 02, 2004 to Aprl 30, 2012 (t,1y) (t,2y) VSTOXX 12M VSTOXX 24M (t,1y) ** 0.25 ** 0.19 ** (t,2y) ** 0.15 ** VSTOXX 12M ** VSTOXX 24M 1 Panel B: January 02, 2004 to July 31, 2007 (t,1y) (t,2y) VSTOXX 12M VSTOXX 24M (t,1y) ** 0.06 * 0.06 * (t,2y) ** 0.07 * VSTOXX 12M ** VSTOXX 24M 1 Panel C: August 01, 2007 to Aprl 30, 2012 (t,1y) (t,2y) VSTOXX 12M VSTOXX 24M (t,1y) ** 0.31 ** 0.23 ** (t,2y) ** 0.18 ** VSTOXX 12M ** VSTOXX 24M 1 Note: One and two astersks denote statstcal sgnfcance at the 5% and 1% sgnfcance level, respectvely. 23

25 Table 7 reports the Ch-squared statstcs and p-values from the VAR Granger Causalty/Block Exogenety Wald tests performed on a VAR(2) system. Ths s the lag length suggested by the Schwarz s nformaton crteron. The absence of autocorrelaton n the resdual terms suggests that no longer lags for the varables are needed. Results show evdence of Granger causalty runnng from the fxed-ncome to the equty market, whereas the reverse does not hold. Ths mples that the changes n (t,1y) mght be used by portfolo managers to mprove VSTOXX 12M forecasts. Table 7. VAR Granger Causalty/Block Exogenety Wald tests between (t,1y) and VSTOXX 12M based on a VAR(2) model Null hypothess Ch-sq statstc Probablty Δl(t,1Y) does not Granger cause ΔlVSTOXX 12M ΔlVSTOXX 12M does not Granger cause Δl(t,1Y) Note: l ( t,1 Y ) and lvstoxx12m t stand for the frst log-dfferences of (t,1y) and VSTOXX 12M, respectvely. To the best of our knowledge, ths s the frst study that examnes whether there s a spllover of mpled volatlty between stock return and nterest rate volatlty ndces wthn the same area. Thus, t contrbutes to the exstng lterature on mpled volatlty spllovers across equty markets (see e.g., Nkknen and Sahlström, 2004; Äjö, 2008; Konstantnd et al., 2008; Kumar, 2012; Sropoulos and Fassas, 2012). 6. Summary and conclusons We suggest for the frst tme a methodology for the constructon of a set of nterest rate volatlty ndces for the Eurozone (s) based on the mpled volatlty quotes of one of the most lqud fxed-ncome dervatves: caps (floors). These ndces reflect the market estmate of the volatlty of three- and sx-month tenor forward rates over dfferent fxed horzons one, two, three, four, fve, seven and ten years. s are constructed through a two-step process. Frst, we apply a strppng procedure consstng of recoverng the mpled volatltes of the ndvdual caplets (floorlets) that compose caps (floors), as these are the contracts that do have an underlyng specfc forward rate. Second, mpled volatltes of near-the-money caplets 24

26 (floorlets) are lnearly nterpolated. Thus, each (t,t ) reflects the mpled volatlty of a theoretcal ATM caplet (floorlet) wth a constant tme to maturty, from t to T. The ATM mpled volatlty wth expry at tme T has a specfc theoretcal nterpretaton: t approxmates the volatlty swap rate (.e., the condtonal rsk-neutral expectaton of the future realzed volatlty of the underlyng asset over the perod [t, T]). Volatlty ndces n the equty markets are referred to as nvestors gauges of fear because they usually spke n perods of fnancal turmol. In ths paper, we extend the emprcal evdence by analyzng the effect of the recent fnancal crss on short- and long-term s. We fnd that the crss has had a deep and lastng effect on nvestors short-term expectatons of volatlty n the fxed-ncome market by Aprl 2012, volatlty levels are more than three-fold the average pre-crss value. More nterestngly, we also fnd that as the crss deepened, t also eventually affected expectatons of volatlty fve- and ten-years ahead the ndces ntate an upward trend n The frst fndng seems to support the nterpretaton of as nvestors gauge of fear for the fxed-ncome market, whereas the second one mght be nterpreted as a sgnal that nvestors foresee long perods of turbulence n nterest rate markets. In addton, we compare the effect of the fnancal turmol on nvestors expectatons of volatlty of nterest rates and stock returns over the next one and two years by usng two equty market volatlty ndces, VSTOXX 12M and VSTOXX 24M. We observe that fxed-ncome and equty market volatlty ndces depct a qute smlar pattern over most of the sample. However, VSTOXX sub-ndces exhbt a lower rse than one- and two-year s durng the crss perod and the sze of the spkes s also smaller. Moreover, by Aprl 2012, VSTOXX sub-ndces are approxmately 30%, whereas the average pre-crss level of the ndces s approxmately 20%. Ths fndng suggests that s have played a greater role as nvestors gauge of fear durng the recent fnancal crss than VSTOXX sub-ndces. Based on these fndngs, we construct a measure of global fnancal rsk averson as a lnear combnaton of nterest rate and stock return volatlty ndces and we show that t moves closely together wth Euro area government yeld spreads from 2009 up to md Fnally, we show that changes n short-term s and VSTOXX sub-ndces 25